HW Sec $5\mathrm{.}1$ Approximating Area: Problem $4$

$(1$ point$)$

In this problem you will calculate the area between $f(x)=4x+8$ and the $x-$axis over the interval $0,2$ using a limit of rightendpoint Riemann sums:

Area $=\underset{n}{lim}({\displaystyle \underset{k=1}{\overset{n}{}}}f(x_k)x)$

Express the following quantities in terms of $n,$ the number of rectangles in the Riemann sum, and $k,$ the index for the rectangles in the Riemann sum.

a$.$ We start by subdividing $0,2$ into $n$ equal width subintervals $[x_0,x_1],[x_1,x_2],$dots,$[x_{n-1},x_n]$ each of width $x.$ Express the width of each subinterval $x$ in terms of the number of subintervals $n.$

$x=$

$$

b$.$ Find the right endpoints $x_1,x_2,x_3$ of the first, second, and third subintervals $x_0,x_1$

$x_1,x_2$

$x_2,x_3$ and express your answers in terms of $n.$

$x_1,x_2,x_3=,(Enter$ a comma separated list.$)$

c$.$ Find a general expression for the right endpoint $x_k$ of the $k^{th}$ subinterval $x_{k-1},x_k,$ where $1kn.$ Express your answer in terms of $k$ and $n.$

$x_k=$

$$

d$.$ Find $f(x_k)$ in terms of $k$ and $n.$

$f(x_k)=$

$$